129 problems

Let $\mathcal{M}=(E,\mathcal{C})$ be an oriented matroid of rank $r$, let $e\in E$, and let $\mathcal{C}^+$ denote its positive circuits. An $e$-positive circuit is a positive circ…

3-geodesic conjecture. For every $d$-dimensional Dantzig figure $(P,x,y)$,

Strong Dantzig conjecture. Every fundamental deformation of a $d$-dimensional Dantzig figure is good; equivalently,

Dantzig conjecture. The graph $E(d)$ is strongly connected: for any vertices $x,y$ of $E(d)$, there is an oriented path from $x$ to $y$.

Face-structure conjectures for tropical polytopes. The following assertions hold: (1) $k$-faces of tropical polytopes are extreme sets; (2) the topological boundary of a $k$-face i…

Tropical halfspace characterization conjecture. A tropical polytope $P$ is pure and full dimensional if and only if it has a halfspace description whose apices are in general posit…

Tropical halfspace representation conjecture. If $P$ is pure, then the halfspaces from a generic lift $\overline{P}$ of $P$ map to tropical halfspaces whose intersection is $P$ its…

Combinatorial characterization conjecture. If the assignment $M\mapsto G_M$ satisfies 1. for every such decomposition,

Let $\mathrm{met}_n$ be the metric polytope, and let its fractional vertices be the vertices that are not integral. The restriction of $\mathrm{met}_n$ to its fractional vertices i…

Partition characterization. Every minimal halfspace with respect to ${\mathcal P}$ has the form

A tropical $d$-plane in $n$-space is a tropical linear space of dimension $d$ in $n$-space. Let $i$ be a face dimension, and map the space to the quotient by the diagonal line: … A…

Collapsing conjecture. Suppose that

Let $i$ be the number of faces of an irreducible $i$-hedrite, and let a central circuit be a central circuit of the hedrite. An irreducible $i$-hedrite is maximal irreducible when…

Oriented-multicut adjacency conjecture. (i) An oriented multicut $\delta'(A)$ is not adjacent to every oriented multicut $\delta'(B)$ such that $B<A^T$. (ii) The orbit represented…

Oriented-multicut facet conjecture. The following inequalities define facets of $OMCUT_n$: (i) zero-extensions of every facet of $OMCUT_{n-1}$, including oriented triangle and non-…

Oriented-multicut skeleton conjecture. (i) All oriented multicuts are extreme rays of $QMET_n$. (ii) The oriented cuts form a dominating clique in $G_{OMCUT_n}$; consequently, the…

Adjacency and diameter conjecture. (i) A triangle facet is adjacent to a facet if and only if they are non-conflicting. (ii) The non-negativity facets $NN_{ij}$ and $NN_{i'j'}$ are…

Let $NHM_n^m$ be the cone of nonnegative $m$-hemimetrics. Denote by $T_{i_1\dots i_{m+1},i_{m+2}}$ an $(m+1)$-simplex facet and by $N_{i_1\dots i_{m+1}}$ a nonnegativity facet. Let…

Let $HM_n^m$, $NHM_n^m$, and $P_n^m$ be the corresponding cones of $m$-hemimetrics, and let the ridge graph of a cone have the facets as vertices, with adjacency when two facets me…

Let $V_n$ be the ground set, and let $\alpha(S_1,\dots,S_{m+1})$ and $\alpha(T_1,\dots,T_{m+1})$ be two partition $m$-hemimetrics on $V_n$. Two rays are adjacent when they generate…

Let $I\subset k[x_1,\dots,x_n]$ be a square-free monomial ideal generated in degree $d$ and having a linear presentation. Let $G_I$ be the graph whose vertices are the minimal mono…

Let ${\mathcal G}'(k,n)$ be the collection of graphs whose vertices are labeled by $k$-subsets of an $n$-element set, with the property that for vertices labeled by $S$ and $T$, th…

Symmetric capacity conjecture. The symmetric capacity satisfies

For each integer $d\geq 3$, let $\mathfrak P_d\subseteq\mathbb R^{2d-3}$ be the integral polytope whose integer points correspond to irreducible Ferrers-diagram pairs…

Mihail–Vazirani conjecture. Every $0/1$ polytope $P$ satisfies